Question

a city council consists of eight democrats and eight republicans. if a committee of four people is selected, find the probability of selecting two democrats and two republicans.
(type a fraction. simplify your answer.)

Explanation:

Step1: Calculate total number of ways to select 4 people from 16 (8 D + 8 R).

The combination formula is \( C(n, k) = \frac{n!}{k!(n - k)!} \). So total ways \( C(16, 4) = \frac{16!}{4!12!} = \frac{16\times15\times14\times13}{4\times3\times2\times1} = 1820 \).

Step2: Calculate ways to select 2 Democrats from 8.

Using combination, \( C(8, 2) = \frac{8!}{2!6!} = \frac{8\times7}{2\times1} = 28 \).

Step3: Calculate ways to select 2 Republicans from 8.

Similarly, \( C(8, 2) = 28 \) (same as Democrats since there are 8 Republicans too).

Step4: Calculate favorable ways (2 D and 2 R).

Multiply the two combinations: \( 28\times28 = 784 \).

Step5: Calculate probability.

Probability \( P = \frac{\text{favorable ways}}{\text{total ways}} = \frac{784}{1820} \). Simplify by dividing numerator and denominator by 4: \( \frac{196}{455} \), then by 7: \( \frac{28}{65} \). Wait, wait, 784 ÷ 28 = 28, 1820 ÷ 28 = 65? Wait, 28×65 = 1820? 28×60=1680, 28×5=140, 1680+140=1820. Yes. And 784 ÷28=28? Wait 28×28=784. So 784/1820 = (28×28)/(28×65) = 28/65. Wait, no: 784 ÷ 28 = 28? Wait 28×28=784, 1820 ÷28=65. So yes, 28/65. Wait, let's check the simplification again. 784 and 1820: GCD of 784 and 1820. Let's find GCD(784,1820). 1820 ÷784=2 with remainder 252. 784 ÷252=3 with remainder 28. 252 ÷28=9 with remainder 0. So GCD is 28. So 784 ÷28=28, 1820 ÷28=65. So 28/65.

Answer:

\(\frac{28}{65}\)